Next we wish to apply the two-time conditional probability to the extended Wigner's friend (EWF) scenario introduced in [1]}. As shown in Figure REF , the EWF scenario consists two space-like separated laboratories \(L_1\)  and \(L_2\) . Each laboratory contains half of an entangled pair of systems \(s_1\)  and \(s_2\) . \(L_1\)  also contains a friend Charlie who can perform measurements on \(s_1\) . Outside \(L_1\)  there is a super-observer Alice who can perform different types of measurements on \(L_1\)  as a whole. Similarly, there is a friend Debbie in \(L_2\) , and a super-observer Bob outside \(L_2\) . Here, we need to consider four POVM measurements, represented by POVM elements \(A, B, C, D\) . Here operators \(A\)  and \(C\)  act on Hilbert space \(\mathcal {H}_{L_1}\) , and \(B\)  and \(D\)  act on Hilbert space \(\mathcal {H}_{L_2}\) . We drop the subscripts of the operators and \(\rho _0\)  for simplifying notations. In a typical EWF experiment, the chosen operators are not all commutative with one another. Specifically, \([A, C]\ne 0\)  and \([B, D]\ne 0\) , while \([C\otimes I_{L_2}, I_{L_1}\otimes D]=0\)  and \([A\otimes I_{L_2}, I_{L_1}\otimes B]=0\) . The two-time probability formulation to compute the conditional probability is needed because the measurements \(C\)  and \(D\)  are taken place before the measurements of \(A\)  and \(B\) . Since \([C\otimes I_{L_2}, I_{L_1}\otimes D]=0\)  and \([A\otimes I_{L_2}, I_{L_1}\otimes B]=0\) , we can assume measurement \(C\)  and \(D\)  are taken place at the same time, \(t_a\) , as \(C\otimes D\)  while measurements of \(A\)  and \(B\)  at a same later time \(t_b\)  as \(A\otimes B\) . Without loss of clarity, we drop the symbol \(\otimes \)  hereafter. Then the conditional probability for \(t_a < t_1 < t_b < t_2\)  is given by
\(\begin{split}&p(ab \text{ }at\text{ } t_2|cd \text{ }at\text{ }t_1, \rho , xy) \\ &=\frac{Tr(\rho \sqrt{C}\sqrt{D}U^{\dag }ABU\sqrt{C}\sqrt{D})}{Tr(\rho CD)},\end{split}\) 
